Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the question, if legitimate at all, should have been restricted to interesting manifestations of a hyperbolic-parabolic-elliptic subdivision, then I can fully agree (although part of the idea was to interpret the question as you see fit); I left it open ended primarily because of the Weil trichotomy, which is of completely different kind and is so much more than a hierarchy, and in relation to which I was interested in hearing other people's opinions and elaborations. See, for instance, how Edward Frenkel, in a fascinating Bourbaki talk, builds upon the Weil trichotomy to introduce a parallel between Langlands and electro-magnetic dualities, which he uses as a springboard for the ideas from physics that have entered the arena of the geometric Langlands program. Or take the cherished three-sided parallel between the basic three-dimensional (from the point of view of etale cohomology) objects and their branched coverings: $\mathbb{P}_{\mathbb{F}_q}^1$, $\mathrm{Spec}(\mathbb{Z})$, and $S^3$, with primes in number fields corresponding to knots in threefolds, $\log{p}$ corresponding to hyperbolic length, etcetera.

To those who were not convinced that there was a neat trichotomy of algebraic surfaces (arguing they should instead form a tetrachotomy by Kodaira dimension), let alone in higher dimensional algebraic geometry, I refer to Sándor Kovács's answer here, which demonstrates rather eloquently the fundamental trichotomy of birational geometry:

Original post. For many purposes, notably in classification hierarchies or in Weil's "big picture" of the fundamental unity in mathematics, it seems as if mathematical reality is more accurately captured by trichotomies than by two-sided dictionaries or questions of "either/or." The most basic is of course the trichotomy negative - zero - positve embodied by the complete ordered field $(\mathbb{R},<)$ --- this is the Arrow of Time, if you will, or the conditioning of a dynamical system into states of past/present/future. As evidenced by some of the examples below, this trichotomy underlies varied, if crude, classification schemes in mathematics.

Other trichotomies arise from closer examinations of a mathematical parallel. Mathematicians have always been fond of discovery by analogy; they take very seriously the intuitions supplied by different yet loosely connected fields. In doing so, they are guided by a tacit, platonic belief in the fundamental Unity of mathematics. An example is the similarity between finite geometries and Riemann surfaces. To explain this parallel, indeed to make sense of it, it is necessary to provide a "middle column" in the dictionary: the arithmetic geometry of number fields and arithmetic surfaces. This leads to the trichotomy that Weil explained so lucidly in a letter (which he wrote in 1940 in prison for his refusal to serve in the army) to his sister, the philosopher Simone Weil. This point of view led, as we know, to an entire new field of mathematical inquiry.

Below I have listed some other cherished mathematical trichotomies. I am interested in seeing yet others, perhaps more specialized. This is my question: add more trichotomies to the list. Furthermore, I am interested in any reflections anyone might have, such as pertaining, for instance, to any of the following questions. Is 3 the most ubiquitous number in coarse classification schemes? Is it fair to say that a given trichotomy echoes the primeval trichotomy $(-,0,+)$? In a given trichotomy, is there a natural "middle column" of a corresponding three-sided dictionary? Is this "middle column" in any way the most fundamental, the most interesting, or the most elusive?

Trichotomies in mathematics: some examples.

The fabric of topology, geometry, and analysis is the real line $\mathbb{R}$. Tarski's eight axioms characterize it in terms of a complete binary total order <, a binary operation +, and a constant 1. (Multiplication comes afterwards - it is implied by Tarski's axioms - and so does the Bourbaki definition of the reals as the complete ordered field). The sign trichotomies $(<,=,>)$ and $(-,0,+)$ ensuing from those axioms have repercussions throughout all of mathematics.

For example, there are three constant curvature spaces, leading to the three maximally symmetric geometries: hyperbolic, flat (or Euclidean), and elliptic (e.g. spherical forms).

In complex analysis, there are three simply connected cloths: the Riemann surfaces $\Delta$, $\mathbb{C}$, and $\hat{\mathbb{C}}$.

The connected component of the group of conformal automorphisms of a compact Riemann surface is one of the following three: trivial, $S^1 \times S^1$, $\mathrm{PGL}_2(\mathbb{C})$.

The complexity of fundamental groups, as showcased first of all by topological surfaces: genuinely non-abelian (perhaps we could say: anabelian) - abelian (or more generally, containing a finite index nilpotent subgroup) - and trivial (or more generally, finite). This is of course related to the subject of growth of finitely generated groups, brought forward by Lee Mosher's answer.

In dynamics, a fixed point (or a periodic cycle) can be either repelling, indifferent, or attracting.

In Thurston's work on surface homeomorphisms, elements of the mapping class group are classified according to dynamics into three types: pseudo-Anosov, reducible, and finite-order.

In algebraic geometry, the positivity of the canonical bundle is central to the classification and minimal model problems. More generally, positivity is a salient feature of algebraic geometry. For a delightful discussion, see Kollar's review of Lazarsfeld's book "Positivity in algebraic geometry." (Bull. AMS, vol. 43, no. 2, pp. 279-284). The most basic example is the trichotomy of algebraic curves (rational, elliptic, general type).

In birational algebraic geometry, at a very coarse level, there are three kinds of varieties out of which a general variety is made: rational curves, Calabi-Yau manifolds, and varieties of general type (or hyperbolic type, if you prefer). For example, an algebraic surface either: 1) admits a pencil of rational curves; or 2) admits a pencil of elliptic curves or is abelian or K3 (or a double quotient of a K3); or else 3) it is of general type. Abelian and K3 are examples of Calabi-Yau manifolds.

More concretely, consider smooth hypersurfaces $X \subset \mathbb{P}^n$. They divide into three types, according to how their degree $d$ compares with the dimension. If $d \leq n$, they contain plenty of rational curves (certainly uncountably many). If $d = n+1$, they are an example of a Calabi-Yau manifold, and typically contain a countably infinite number of rational curves. (The generating function of the number of rational curves of a given degree is then a very interesting function, of significance in the physics of quantum gravity.) And if $d \geq n+2$, then $X$ is of general type, and it is conjectured to typically contain only finitely many rational curves. (More precisely, Bombieri and Lang have conjectured that a variety of general type contains only finitely many maximal subvarieties not of general type).

In diophantine geometry, rational points are supposed to come from rational curves and abelian varieties. The sporadic examples are believed to be finitely many. This leads to the following trichotomy for the growth rate of the number of rational points of bounded (big, i.e. exponential) height: polynomial growth - logarithmic growth - $O(1)$. Furthermore, even in dimension 1, it is for abelian varieties that the situation is the deepest and the most mysterious.

In topology, it seems as if the interesting dimensions fall into three qualitatively different ranges: $d = 3$, $d = 4$, and $d \geq 5$. (Although this might be stretching it a bit too much). Of these, four dimensions -- the "middle column" -- is the most mysterious, and also the most relevant for physics.

The "Weil trichotomy," of course, goes at least as far back to Kronecker and Dedekind: curves over $\mathbb{F}_q$ - number fields - Riemann surfaces. Class field theory and Iwasawa theory are particularly eloquent examples of this trichotomy. Another example is of course the zeta function and the Riemann hypothesis.

One would be tempted to extend the latter trichotomy to [non-Archimedean world ($p$-adic, profinite) - global arithmetic - Archimedean world (geometry, topology, complex variables)], if the middle column did not subsume (much of) the flanking columns. Likewise the triple [$l$-adic cohomology-motive-Hodge structure] would probably not be admissible.
Here is a variation on the theme (you may find it to be rubbish, in which case throw it away). There are two ways of completing (or taking limits of) the regular polygons $C_n$. The first is to think of $C_n$ as $\frac{1}{n}\mathbb{Z}/\mathbb{Z}$ and take the direct limit (in this case, union, or synthesis: $\rightarrow$), which is $\mathbb{Q}/\mathbb{Z}$. Completing, we get the circle $S^1 = \mathbb{R}/\mathbb{Z}$, which is the simplest manifold. The second is to think of $C_n$ as $\mathbb{Z}/n$ and take the projective limit (or deconstruction: $\leftarrow$), which is $\hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p$, the profinite version of the circle. In this way, Archimedean (continuous) objects and $p$-adic objects may be seen as the two possible different limits (synthesis and deconstruction) of the same finite objects. Taking $C_n$ to be more general finite groups, we get essentially all the Lie groups, on the one hand; and all the profinite groups, on the other hand.

That we live in three perceptible spatial dimensions does not, of course, fit our bill. But in 1984, Manin published an article ("New dimensions in geometry") in which, guided by ideas from number theory (Arakelov geometry) and physics (supersymmetry), he proposed that there are three kinds of geometric dimensions, modeled on the affine superscheme $\mathrm{Spec} \mathbb{Z}[x_i;\xi_j]$, an "object of the category of topological spaces locally ringed by a sheaf of $\mathbb{Z}/2$-graded supercommutative rings." Here, $\xi_j$ are "odd," anticommuting variables, commuting with the "even" variables $x_i$. See the three coordinate axes $x, \xi$ and $\mathrm{Spec} \mathbb{Z}$ in his picture of "three-space-2000." The arithmetic axis $\mathrm{Spec} \mathbb{Z}$ is implicit in complex algebraic geometry, and is essential in problems such as the Ax-Grothendieck theorem and the construction of rational curves in Fano manifolds.

Zeta functions can by dynamical (Artin-Mazur); arithmetical on schemes of finite type over $\mathbb{Z}$ (Riemann and Hasse-Weil); and geometric (Selberg's zeta function of a hyperbolic surface).

In Model theory, there is an important trichotomy between super-stable theories, strict-stable (stable but not superstable) theories, and non stable theories.

It seems fair to say that there are three kinds of three-dimensional simply connected spaces: $\mathbb{P}_{\mathbb{F}_q}^1$, $\mathbb{Spec}(\mathbb{Z})$ compactified at archimedean infinity, and $S^3$. This brings about the Mazur knotty dictionary and the fruitful analogy between primes and knots (especially hyperbolic knots).

A finitely-generated infinite group has 1, 2, or infinitely many ends. (But if you leave out the word "infinite", it becomes a tetrachotomy; this seems to be a weakness of the question.)
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Artie Prendergast-SmithFeb 2 '13 at 21:14

13

From the movie $\pi$: "Hold on. You have to slow down. You're losing it. You have to take a breath. Listen to yourself. You're connecting a computer bug I had with a computer bug you might have had and some religious hogwash. You want to find the number 216 in the world, you will be able to find it everywhere. 216 steps from a mere street corner to your front door. 216 seconds you spend riding on the elevator. When your mind becomes obsessed with anything, you will filter everything else out and find that thing everywhere."
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Benjamin DickmanFeb 3 '13 at 10:05

31

This feels like an awfully arbitrary and contentless question to me. This seems like asking, what are some interesting sets with 3 elements? Voting to close. A more reasonable question would be to ask for examples where there is an interesting hyperbolic-parabolic-elliptic trichotomy and connections between these examples.
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Eric WofseyFeb 3 '13 at 13:28

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Let's please not be quick to close this question. There are some intriguing clusters of answers, some along a (negative, 0, positive) axis, and others along a (closed, open, closed) axis [if I may put it crudely], and others along other axes still. There is a certain amount of erudition in both the question and answers, and there is some potential to glimpse "analogies between analogies" in the words of Ulam. It could be a useful exercise to pursue this.
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Todd Trimble♦Feb 4 '13 at 2:29

31 Answers
31

Almost everything in mathematics, indeed, can be "hyperbolic", "parabolic" or "elliptic".
Like PDE's, Riemann surfaces, or manifolds of higher dimension, fractional-linear transformations, fixed points of a map, etc.

Not even mentioning the
3 kinds of the conic sections:-)

Of course this can be traced back to the fundamental trichotomy "positive", "zero" and "negative".

In differential geometry we have three great areas: "positive curvature",
"negative curvature" and
"zero curvature".

The trichotomy elliptic-parabolic-hyperbolic holds for Riemann surfaces, but in higher dimensions there are far more possibilities. In dimension 3, there are Thurston's 8 geometries ( and usually 3-manifolds have to be decomposed into pieces to be geometric). In dimension 4, Wallach had a list of 17 geometries, but one item in his list actually contains an infinite number of geometries. (And certainly not all 4-manifolds are locally homogeneous.)
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ThiKuFeb 4 '13 at 12:31

3

Of course, in higher dimensions we have more possibilities, but still they are frequently roughly classified into "hyperbolic", "parabolic" and "elliptic" types.
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Alexandre EremenkoFeb 5 '13 at 12:52

+1. I think your example is good, and indeed does resonate with these other trichotomies, in fact builds a bridge between "negative, zero, positive" and "false, between-false-and-true, true" that I hadn't guessed. Indeed, it should be compared to the MO question on why we use the open-set formulation of 'topology', especially in answers such as sigfpe's (Dan Piponi's) and the summary given by Marcos here: mathoverflow.net/questions/19152/…. Compare also Tarski's interpretation of intuitionistic logic in a topology of a space.
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Todd Trimble♦Feb 4 '13 at 10:32

After passing to a subsequence if necessary, a sequence of real numbers either (a) converges to a real number; (b) diverges to $+\infty$; or (c) diverges to $-\infty$. In a similar vein, a sequence of positive real numbers either (a) converges to a positive real; (b) diverges to $+\infty$; or (c) diverges to $0$. In nonstandard analysis, these trichotomies become those of being bounded, negative unbounded, or positive unbounded, or of being infinitesimal, unbounded, or neither (i.e. both bounded, and bounded away from zero). These are of course variants of the basic $(-,0,+)$ trichotomy.

Up to isomorphism, there are only three types of (connected) one-dimensional algebraic groups over an algebraically closed field: the additive group of the field, the multiplicative group of that field, and the elliptic curves over that field. (This last family would definitely be the "middle column".) This is of course connected to many of the other trichotomies previously mentioned. On the Riemann surface side, it comes from the fact that all one-dimensional connected complex groups are isomorphic to ${\bf C}/\Gamma$ for some discrete subgroup $\Gamma$ of ${\bf C}$, which can have rank 0 (additive case), 1 (multiplicative case), or 2 (elliptic curve case).

If one squints at it in just the right way, the classification of finite simple groups is a trichotomy: cyclic, Lie type (including Lie over F_1, i.e. alternating group), or sporadic. (Of course, it can be sliced in many other ways; counting the items in this Wikipedia list, for instance, would make it a tetratetracontachotomy.) Sometimes it is conceptually useful to split up the large Lie type groups into three regimes: large characteristic but bounded rank; large rank but bounded characteristic (including the alternating groups); and large characteristic and large rank. (Alternatively, one can partition into bounded rank, alternating, and unbounded rank.) One can debate as to which of these categories is the "middle column".

If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the Littlewood-Paley trichotomy: (a) "high-low" interactions with $|\xi_1| \gg |\xi_2|$ and $|\xi_1| \sim |\xi_3|$; (b) "low-high" interactions with $|\xi_1| \ll |\xi_2|$ and $|\xi_2| \sim |\xi_3|$; and (c) "high-high" interactions with $|\xi_1| \sim |\xi_2|$ and $|\xi_1| \gg |\xi_3|$. (One has to carefully demarcate the boundaries between these three possibilities to ensure it is a true trichotomy.) To an algebraic geometer, this would reflect the Y-shaped nature of the amoeba of the set $\{ (\xi_1,\xi_2,\xi_3): \xi_3 = \xi_1 + \xi_2 \}$. This trichotomy is important in harmonic analysis and PDE, and in particular in the paradifferential calculus of products and paraproducts (see e.g. this blog post of mine). Often, one of the three interactions will be the most dominant, reflecting either a high-to-low frequency cascade or a low-to-high frequency cascade, but it depends heavily on the situation. Note that this trichotomy is basically a variant of the $(<,=,>)$ trichotomy.

(ADDED LATER) Another variant of the $(<,=,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE problem, such as an initial value problem in a certain function space) can be classified as subcritical, critical, or supercritical, depending on whether the nonlinear component of the PDE is "weaker than", "comparable to", or "stronger than" the linear component in a suitable asymptotic limit (usually the fine scale/high frequency limit, although for scattering theory the coarse scale/low frequency limit is the relevant one instead). This distinction (which can usually be made precise through a scaling analysis or dimensional analysis) is often decisive in determining the difficulty level of the PDE problem. For instance, the regularity problem for 3D Navier-Stokes is supercritical and thus considered close to intractable, but 2D Navier-Stokes is critical and was solved decades ago. The global analysis of Ricci flow (with surgery) was considered supercritical until Perelman discovered new monotone quantities that made it critical, which was absolutely necessary for Perelman to be able to execute the rest of Hamilton's program and solve the Poincare and geometrisation conjectures. In this trichotomy, the critical (or scale-invariant) case is generally viewed as the most interesting and delicate, with some very nice mathematical tools coming into play to control the interaction between different scales. Perhaps it should also be pointed out that this trichotomy is orthogonal to the elliptic/parabolic/hyperbolic trichotomy, which only concerns the linear component of the PDE and not the nonlinear component, and all nine combinations (critical elliptic PDE, supercritical parabolic PDE, etc.) are studied in the literature.

(ADDED YET LATER) In analysis, there are basically three scenarios that prevent a weakly convergent sequence $f_n$ of functions in some function space from being strongly convergent in that space: (a) escape to "horizontal infinity" (basically, the support of the function runs off to spatial infinity, i.e. moving bump type examples); (b) escape to "vertical infinity" (the peaks of the function go to infinity, e.g. a sequence of approximations to the identity converging weakly but not strongly to a delta function); and
(c) escape to "frequency infinity" (the functions become increasingly oscillatory). If one can shut down all three modes of escape then one can recover strong convergence, and thus also strong (pre)compactness, cf. the Arzela-Ascoli theorem which has three hypotheses (compact domain, pointwise boundedness, equicontinuity) to shut down (a), (b), and (c) respectively. In Section 2.9 of Lieb-Loss, these three scenarios are called "wanders off to infinity", "goes up the spout", and "oscillates to death" respectively.

+1 for tetratetracontachotomy. I think I typed that correctly.
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Lee MosherFeb 3 '13 at 14:41

1

Perhaps one should mention that with the classification of finite simple groups, things are usually in dichotomies: for instance, the dichotomy theorem (odd or even-type characteristic), small vs. big (quasithin or large), etc.
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Steve DFeb 4 '13 at 4:23

A simple yet useful one: An irreducible complex representation of a compact Lie group is either 'real', 'quaternionic', or 'complex'. That is, it is the complexification of a real irreducible, or it can be considered quaternionic through the existence of an equivariant conjugate-linear (real-)automorphism $j$ that squares to $-I$, or it is neither.

The statement combines Schur's Lemma and the fact that there are three associative real division algebras, here seen through complex eyes.

Arnol'd thought this was a particularly fundamental trichotomy, and he has a fascinating table of "related" trichotomies in his book "Arnol'd's Problems" (sorry for the apostrophe catastrophe). Here's a picture I uploaded a few years ago:

I wonder if these triples are in the same "family" as $(-1 ,0 , 1) $. At first they seem to be different because there is no distinguished element, analogous to $0$ - but perhaps $\mathbb C$ plays this role? I think the proof that there are three division algebras might actually use the $(-1,0,1)$ trichotomy, but I don't have a moment to think about it right now - maybe someone can leave a comment confirming this? (Edit: I could also imagine $\mathbb R$ playing the role of $0$, because it is the identity element for the tensor product, for example - any opinions either way?)

Another question I find interesting, which I'd hoped to think about at some point for fun, asks not just for (related) occurrences of 3-element sets in mathematical classifications, but more generally, for related occurrences of $n$-element sets for small $n$. I'd always dreamed of taking the number $5$ of regular platonic solids, for example, and trying to deduce from this as many other "small, finite number classifications" as possible.

@Sam, in the setting of my answer, if you integrate $\chi(g^2)$ over all $g$ in the group $G$ where $\chi$ is the character of the rep, you get $1,0,-1$ when the rep is 'real', 'complex' or 'quaternionic' respectively. Another manifestation of the $1,0,-1$ trichotomy is in the behaviour of tensor products over $\mathbb{C}$ of the different types.
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Paul ReynoldsFeb 3 '13 at 14:12

Those are the three distinctions we have for cardinalities. For most people uncountable would usually mean $2^{\aleph_0}$. But even for set theorists, given a model of ZFC, the finite sets are finite, the countable sets are countable, and the rest is madness$^*$.

Replacing cardinality by topological-measure theoretic properties of subsets of an ordinal $\kappa$, there are non-stationary sets (small); stationary sets (big, but not too big); and clubs (which is practically everything).

An infinite finitely generated group $G$ has either 1, 2 or infinitely many ends; the last case is equivalent to saying that $G$ splits as an amalgamated free product or an HNN extension over a finite subgroup, by a theorem of Stallings. (Of course the assumption that $G$ is infinite brushes a fourth possibility under the carpet, namely that the number of ends is zero.)

Every isometry of a proper $CAT(-1)$ space is either elliptic, parabolic, or hyperbolic: elliptic means fixes a finite point; hyperbolic means fixes two infinite points connected by a translation axis, equivalently translation distance bounded away from zero; parabolic means translation distance limiting to zero but no fixed point. There are versions for proper Gromov hyperbolic spaces, and even for the nonproper case, if you are willing to "quasify" the statements of the cases, and if you are willing to let the trichotomy degenerate to a dichotomy.

Every isometry of Teichmuller space is either elliptic, parabolic, or hyperbolic. This is Bers' form of Thurston's trichotomy for mapping classes: finite order, reducible, pseudo-Anosov. This trichotomy also has an interpretation in terms of the action of the mapping class group on the curve complex which is a nonproper Gromov hyperbolic space by a theorem of Masur and Minsky.

For elements of $Out(F_n)$, the outer automorphism group of a rank $n$ free group, there are related trichotomies and other -otomies coming from the work of Bestvina, Feighn, and Handel on relative train track theory. The simplest one is that every element of $Out(F_n)$ is either of finite order, or of polynomial growth, or of exponential growth.

The reduction (special fiber) $E_{\mathfrak p}$ of (the Neron model of) an elliptic curve $E$ modulo a prime ${\mathfrak p}$ is one of:

good reduction = stable reduction = $E_{\mathfrak p}$ is non-singular

multiplicative reduction = semi-stable reduction = $E_{\mathfrak p}$ is a product of the multiplicative group times a finite group

additive reduction = unstable reduction = $E_{\mathfrak p}$ is a product of the additive group times a finite group

Of course, this trichotomy is a reflection of the fact that there are only three sorts of connected one-dimensional Lie groups, namely the additive group, the multiplicative group, and the compact case (elliptic curves).

Every finitely generated group is either of polynomial growth, intermediate growth, or exponential growth.

As a statement, there is not much to this, the only mathematical content is that the growth function of every finitely generated group has an exponential upper bound.

But as a method of classifying finitely generated groups, it has been very fruitful: Gromov's theorem on groups of polynomial growth; the incredibly rich theory that arose from Grigorchuk's original construction of an intermediate growth group; and the emergence of rich classes of exponential growth groups such as word hyperbolic groups.

Let me point out that Vladimir Arnold was quite interested in similar question.
He called subj. "mathematical trinities", see e.g. his paper "Symplectization, Complexification and Mathematical Trinities". As far as I remember from his lectures, his ideas
were that many of these "trinities" are actually related to each other; and he also
considered subj. as a tool to invent to theories: see question marks at already cited
"Arnold's table": jpg.

Let me also mention some "trinities" which occur in my own research related to Capelli identities (which are some non-commutative analogs of det(AB)=det(A)det(B) ).

Matrix trinity - a) generic b) symmetric c) antisymmetric

Here how it goes in Capelli (and related Cayley) identities:

a) generic matrices - original Capelli identity has been discovered by Capelli in 19-th century - it is for "generic matrices" $A=x_{ij}$ $B = \partial_{ji}$

Similar generalization were found for Cayley identity respectively: a) attributed to Cayley
b) Garding 1948 c) Shimura 1984 - see arXiv:1105.6270 for quite a complete information.

My question: is it really trinity ? Or you can propose some analogs of Cayley-Capelli for some other matrices, say "symplectic" ?

It is might be strange, but other trinities like R,C,H also appears
in the Capelli story - and they give different identities.
Moreover trinities can be combined and we might get trinity^trinity^trinity...

Actually H-analog of the Capelli identity is not fully known for the momemnt - only
analog for 1x1 matrices has been discovered quite recently by student of R. Borcherds,
An Huang. Looking at this example I proposed some C-analogs of Capelli identities. Actually all generic/symmetric/antisymmetric can be complexified, hopefully there should exist quaternionic analogs and thus we might have trinity^trinity. Some partial
results of trinity^trinity spirit for Cayley identity contained in loc. cit.

There are certain analogs of Capelli identities for classical Lie algebras:
this can seen as gl/so/su trinity, well probably it is not the trinity in some strict sense. I have no idea can we have something like trinity^trinity^trinity ...

a function of a complex variable with an algebraic addition theorem must be: 1) A rational function, 2) A rational function of e^px, or 3) A rational function of the Weierstrass elliptic function and its derivative.
Trig functions based on convex curves

"there is a trichotomy of curves given by g=0, g=1, and g≥2. If you look at topological, geometric, arithmetic properties of these curves, their properties align very strongly with these classes."
Why should I believe the Mordell Conjecture?

"one of the most amazing facts about logic is that consistency strength boils down to the question "what is the fastest-growing function you can prove total in this logic?" As a result, the consistency of many classes of logics can be linearly ordered! If you have an ordinal notation capable of describing the fastest growing functions your two logics can show total, then you know by trichotomy that either one can prove the consistency of the other, or they are equiconsistent."
http://cstheory.stackexchange.com/questions/4816/axioms-necessary-for-theoretical-computer-science/4821

In thermodynamic formalism for dynamical systems, a Hölder continuous potential function $\phi$ on a countable state topological Markov chain $(X,\sigma)$ is either positive recurrent, null recurrent, or transient. These correspond to the three possibilities for equilibrium states (shift-invariant measures maximising the quantity $h(\mu) + \int_X \phi\\,d\mu$): existence of a finite equilibrium state is equivalent to positive recurrence; null recurrence is the boundary case where the equilibrium state becomes $\sigma$-finite but not finite, and transience is the case where there is no equilibrium state (all the weight has gone to infinity).

These can be characterised in terms of a particular sequence $a_n>0$: positive recurrence is equivalent to $\limsup a_n > 0$, null recurrence is equivalent to $a_n\to 0$ and $\sum a_n=\infty$, and transience is equivalent to $\sum a_n<\infty$. I imagine this trichotomy for sequences appears in other places as well.

Edit: It's worth mentioning that this trichotomy is also true for random walks on directed graphs (weighted or unweighted) -- historically I believe this is where it was first studied and where the terminology came from, but as a dynamicist I more immediately think of the interpretation above. In this setting the interpretations are as follows:

Positive recurrent -- with probability 1, a random walk returns to where it started, and the expected return time is finite.

Null recurrent -- the walk returns to the starting position with probability 1, but the expected return time is infinite.

Transient -- with probability 1, the walk never returns to its starting position.

Over on Riemann surface land, this is the positive curvature / zero curvature / negative curvature trichotomy. (So this is the translate of one trichotomy by another (the Weil trichotomy)!)
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Terry TaoFeb 4 '13 at 16:29

A local domain can be pure characteristic $0$, pure positive characteristic, or mixed characteristic. (Meaning that the algebra has characteristic $0$ but its residue field has positive characteristic.)

Of course, this being a trichotomy relies on the fact that it's a domain; otherwise you could also have characteristic $p^n$ (with the residue field having characteristic $p$). I don't really know how that case is classified (I guess it's a form of mixed characteristic)...

Here is a cluster of examples with a common theme, based partly on comments here and in an MO thread on whether an empty space should be considered connected, and partly on an article in the nLab, "too simple to be simple".

One should note that some of these trichotomies were once-upon-a-time considered dichotomies; for example, for many people in the past, 1 was a prime number. Also one should notice that there are a number of cross-connections (homomorphisms, if you will) between these examples, and that list is by no means complete (this is CW, so feel free to add more!).

A module can be reducible, irreducible, or zero.

A filter in a Boolean algebra can be "submaximal" (faute de mieux!), a maximal filter = ultrafilter, or an improper filter.

An element in a p.i.d. is composite, prime, or a unit.

A topological space (or a graph) can be disconnected, connected, or "unconnected" (empty).

As for elements satisfying a predicate, one can have a multiplicity, unique existence, or nonexistence.

The last trichotomy is often considered a dichotomy: nonuniqueness vs. uniqueness. (I.e., nonexistence falls under the scope of uniqueness = "at most one".) But experience in mathematics, e.g. in category theory and its focus on universal properties, shows that unique-existence deserves to be considered in a category of its own.

Unfortunately, the p.i.d. example has the blemish that one has to exclude the zero element (or add it to make it a tetrachotomy).
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Terry TaoFeb 6 '13 at 18:43

1

Oh, thanks for pointing that out. I guess one could say instead that an ideal in a commutative ring is either submaximal, maximal, or unit, but this makes it look a lot like the previous example (in fact, the previous example becomes a special case in some sense). Perhaps I'll edit; I don't know.
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Todd Trimble♦Feb 6 '13 at 18:52

This trichotomy plays a key role in the study of the geometry of spherical varieties, a class of algebraic varieties that includes grassmannians, flag varieties, toric varieties, algebraic monoids and symmetric spaces. It is particularly important in understanding the analogues of Schubert subvarieties (i.e. closures of orbits of a Borel subgroup) of a spherical variety.

In this example, there is no "middle" case as there is no intrinsic order to the three types.

This is related to the trichotomy of elements of $PGL_2$ which are either elliptic, parabolic, or hyperbolic.
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Ian AgolFeb 3 '13 at 3:36

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So I really should have ordered the list in the more fun manner of N-U-T. As far as I know, there is no use made of the ordering of this trichotomy in the study of spherical varieties, at least not explicitly.
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Michael JoyceFeb 3 '13 at 14:11

Suppose that a group $G$ has an action on a tree with no inversions
and no global fixed point. Then either (a) $G$ is expressible as a
(non-trivial) amalgamated free product; (b) $G$ is indicable (i.e.
it maps onto an infinite cyclic group); or (c) $G$ is expressible as
the union of a strictly ascending sequence of subgroups.

The negation of this property (every action of $G$ on a tree without
inversions has a fixed point) is called FA by (J.-P.) Serre.

A more mundane, but related, example: a single automorphism of a
tree either fixes a point, inverts an edge, or has an invariant
(doubly-infinite) line contained in the tree on which the automorphism acts by
translations.

The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families:
the $sphere$;
the connected sum of $ g $ $tori$, ;
the connected sum of $ k $ $real$ $projective$ $planes $.
this is a simple example of the trichotomy.sphere can be taken as $ 0 $ tori. so $SPHERE$ serves the middle column.

There are three spaces associated to a fibration: the total space, the base space and the fiber.

Here are some examples

Taking the modulus of a real number give an orbifold covering $|-|:{\mathbb{R}}\to {\mathbb{R}}_+$ with fiber ${\mathbb{Z}}/2$. One sheet is the positive reals, the other sheet is the negative reals and the point of ramification is at 0. This recovers the positive, zero, negative trichotomy.

Given a Lawvere theory $T$, consider a model $A:T\to {\mathcal{C}}$ internal to a category ${\mathcal{C}}$ with finite products. A fiber, or more accurately, the fiber product $T\times_{\mathcal{C}} 1$ with the terminal category corresponds to an object of this model. The triple $(T,A, T\times_{\mathcal{C}} 1)$ corresponds to Lawvere's trichotomy of abstract general, concrete general and particular.